Observing the Truth: Diagrams, Sets and Free Rides

Abstract

syntax choices1.

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statements, such as a set of (perhaps a single) Euler diagrams. Such a translation essentially chooses a representation, at the abstract syntax level, of the originally de ned statements in the other notation. Here, this idea is generalized to the notion of an observational advantage that does not require the existence of a translation between notations. Instead, all that is required is for the two sets of statements to be semantically equivalent.

To illustrate, suppose we have the two set-theory statements P = Q and P \ R = ;. The two Euler diagrams d1 and d2 in gure 2 are, between them, semantically equivalent to P = Q and P \ R = ;:

D = fd1; d2g

S = fP = Q; Q \ R = ;g: We can observe P \ R = ; from S because it is written explicitly in S. However, we cannot observe P \ R = ; from either d1 and d2. In this case, from D we must infer d, which directly expresses P \ R = ;. Therefore, S has an observational advantage over D. Moreover, D also has an observational advantage over S, since we can observe Q \ R = ; from D but not S.

Through de ning what it means to be able to observe one statement from a set of statements, we can precisely describe the observational advantages of one set of statements over another. In particular, in the case of Euler diagrams and set theory we can prove that Euler diagrams are observationally complete. That is given a nite set of set-theory statements, S, there exists a single Euler diagram, d, that is semantically equivalent to S such that any set-theory statement that can be inferred from S can be observed from d [ 5 ]2. Thus, it is possible to characterise, formally, the observational advantages of Euler diagrams over settheory. Importantly, the theory of observational advantages allows us to explain why diagrams are advantageous representations of sets in a more general way than free rides.

Acknowledgement This extended abstract is based on research conducted collaboratively with Mateja Jamnik and Atsushi Shimojima [ 5, 6 ]. 2 We note that d must explicitly represent all sets occurring in S and that the inferred set-theory statements must only represent sets occurring in S; we refer the reader to [ 5, 6 ] for more details.

References

1. Blake , A. , Stapleton , G. , Rodgers , P. , Howse , J.: The impact of topological and graphical choices on the perception of Euler diagrams . Information Sciences 330 , 455 { 482 ( 2015 )

2. Rodgers , P. , , Zhang , L. , Purchase , H.: Wellformedness properties in Euler diagrams: Which should be used ? IEEE Transactions on Visualization and Computer Graphics 18 ( 7 ), 1089 { 1100 ( 2012 )

3. Rodgers , P. , Stapleton , G. , Chapman , P. : Visualizing sets with linear diagrams . ACM Transactions on Computer-Human Interaction 22 , 27 :1{ 27 : 39 ( 2015 )

4. Shimojima , A. : Semantic Properties of Diagrams and Their Cognitive Potentials . CSLI Publications , Stanford, CA ( 2015 )

5. Stapleton , G. , Jamnik , M. , Shimojima , A. : What makes an e ective representation of information: A formal account of inferential advantages . Journal of Logic , Language and Information (under review)

6. Stapleton , G. , Jamnik , M. , Shimojima , A. : E ective representation of information: Generalizing free rides . In: Diagrams 2016 . Springer ( 2016 )